On complete monotonicity of certain special functions

Zhang, Ruiming

doi:10.1090/proc/13878

On complete monotonicity of certain special functions
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Proc. Amer. Math. Soc. 146 (2018), 2049-2062 Request permission

Abstract:

Given an entire function $f(z)$ that has only negative zeros, we shall prove that all the functions of type $f^{(m)}(x)/f^{(n)}(x),\ m>n$ are completely monotonic. Examples of this type are given for Laguerre polynomials, ultraspherical polynomials, Jacobi polynomials, Stieltjes-Wigert polynomials, $q$-Laguerre polynomials, Askey-Wilson polynomials, Ramanujan function, $q$-exponential functions, $q$-Bessel functions, Euler’s gamma function, Airy function, modified Bessel functions of the first and the second kind, and the confluent basic hypergeometric series.

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